the-usual-transformations-on-homogeneous-coordinates-for-2-mathrmd-computer-graphics-involve-3-times

Question

The usual transformations on homogeneous coordinates for 2 $\mathrm{D}$ computer graphics involve $3 	imes 3$ matrices of the form $\left[\begin{array}{cc}{A} & {\mathbf{p}} \ {\mathbf{0}^{T}} & {1}\end{array}ight]$ where $A$ is a $2 	imes 2$ matrix and $\mathbf{p}$ is in $\mathbb{R}^{2} .$ Show that such a transformation amounts to a linear transformation on $\mathbb{R}^{2}$ followed by a translation. [Hint: Find an appropriate matrix factorization involving partitioned matrices.]

Ashley Boni · Accepted Answer

so we&#x27;re given that on upon transformation is one of the form t of X is equal to mu tricks, eh? Turns the vector X plus a vector b ah, where a is on by n matrix and B is on and by one vector s So we want to show that if bees non zero the tea&#x27;s not a linear transformation So we need to check the condition for TV linear. So FT was Lanier ah t of X plus y would have to equal t of x plus t of Why so let&#x27;s check both sides. So the left hand side t of explosive by it&#x27;s just equal thio a times that&#x27;s close. Why plus B and T X plus you, Why iss equal to you? A x It must be waas a by plus B. And since these things are not equal to each other, he&#x27;s not letting her transformation. And also another way, Um, it&#x27;s a check. If t was linear than t of zero would have to equal zero. But t of the zero factor is just equal to a times Does your elector plus B which is just the vector B. So if these non zero then t is not

Matthew Markham · Answer

Okay, first transformation T one, We have matrix a Now, to get to find the colonel. We first reduce this in tow R e f, which means for and for a is takes one simple operation we for the second row, we subtract two of the first from 1 to 2, and it gives us one minus 100 Then that we then that means that X might that why isn&#x27;t free variable and X men and, uh, any vector were X minus y equals zero then had then makes, uh, a codas here is on the back of a session. It goes to zero. So this is a colonel space of a Why? Why? Because ax equals why? Yeah, se. And same thing with that being, we need a few more operations. So for the 1st 1 we get, he simply switched easily. Switch these two rows around. Then we subtract the first row or the first road by the second role, giving us one minus 2 to 1. And then, uh, the second road twice attracted means attract twice by the first Roque. And that&#x27;s one minutes to 05 Just by the second by five against one minus two was your one and then subtract the first roll twice by the second giving us the identity matrix. And then that simply means that the Colonel is just going to be the vector zero. Because no other, no other is going to work for them for find the best kernel of t one t two. First we get first, we find 22 just 80 and find the first vector in a B. We get me the time three times the first row and a by the first column and be too time to minus 31 times three and then so on. And then for the second calling, we have one times one plus minus one times minus one equals 200. So long And then we get this matrix and very simple way. Add second row twice by the first, giving us minus 1200 and then me times first road by minus one and it gives us one minus 200 And so that gives 10. So then then then that this is the area. And so we get the equation X minus two white was no. And then this is going to be the colonel to Why? And common line, We&#x27;re wise element. Same thing for T two t one is at this times be a similar operation for the 1st 1 Oh, to times one plus one time to for and then two times minus one plus mice to use my four. So on get way that we then get this matrix and again. Very simple. We switched the out We switched the rose around subtract by the second row by four of the first row giving us this in the artery f and still in ST Same thing for our first. Our first question is X minus Y equals zero And that is why why earlier? And this is the kernel of t two t one.

Ashley Boni · Answer

So we&#x27;re given the mapping t of X equals X with a having columns. A woman, A two which were the victors during the picture to the right. So we want to specifically find the image of the vector. Negative 13 under chief. So we want a plot t of negative one. So this is equal to the matrix? A So a one a two multiplied by the back to rex, which is negative. 13 So multiplying. Um, he&#x27;s out. We get negative long times they won plus three times a two. So we know that the image of this is gonna be the parallelogram formed by these two factors. Um, And then if we connect to the opposite side of the parallelogram from the origin, we get the vector corresponding to the image under the specific director. So let&#x27;s first round negative a one. So we&#x27;re gonna take our vector 81 and we&#x27;re going to flip it in the opposite direction. So we&#x27;re just going to continue this in the negative direction. So this is negative one. So now 382 we&#x27;re just going to scale or vector a two by three. So we&#x27;re gonna make it three times as long as it already is. So let&#x27;s just add another one. So there&#x27;s two way too on here. ISS 32 So, of course, we&#x27;re starting from the origin. So was this entire thing. So now let&#x27;s draw the parallelogram formed by these two vectors. So if I can draw this looks like something like this, and then we&#x27;ll connect parallel to the blue side. So if we want to draw our vector uh, t of negative 13 we&#x27;re gonna connect the origin to the opposite side of parallel occur. So that&#x27;s our vector of the image of negative 13 under the transformation T.

Ashley Boni · Answer

So we want to find the Matrix Tea for a mapping from our TUTTO are too. That leaves the one unchanged and maps to took you two plus two equals. So where do we start? We know that the Matrix has the form t of even one as the first column and t of e to as the second calm. So we want to be want to remain the same. So in other words, we want t of you want to equal one waas zero e too. And we want teeth of e too the equal. You too. Plus three you won. So in order to write this, then matrix form TV once the first column. So it&#x27;s one yuan plus zero e too. And TV too is equal to 31 plus one e two and that is our matrix t.

Rakvi . · Answer

corresponding to t one and t two, that is, we&#x27;re just multi play David Lee. And this is a quartile producto fan. Be which is a little You just multiply the two matrices. Oh, so the fourth century is minus one times one, which is minus one plus minus four, which is minus way. Second injury is minus five Thought entries one and last entry is with Dean. Now, to find out Tito with the one just multiply, baby they so you will get the falling metrics. It is a quartile 14 seven do in the last two entries minus four. So as you can see, these stool compositions are not seem because even because if they were equal, the images of each of the vectors would be equal like they would send. I liked even after you two off We should be quantity to be one off me For every victory in our but the one Pito off one gonna zero. As you can see, it&#x27;s just minus right. Come on, one that is the first column off your matrix. But do you do do you? Enough. The same victor is for the in Goma do which which are different vectors. So they&#x27;re not seem

Problem

Show that the transformation in Exercise 7 is equ…

Problem 13 Easy Difficulty

Answer

$\left[\begin{array}{cc}{I} & {p} \\ {0^{T}} & {1}\end{array}\right] \cdot\left[\begin{array}{cc}{A} & {0} \\ {0^{T}} & {1}\end{array}\right]=\left[\begin{array}{cc}{A} & {p} \\ {0^{T}} & {1}\end{array}\right]$

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Grace H.

Kayleah T.

Maria G.

Kristen K.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Kayleah T.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\left[\begin{array}{cc}{I} & {p} \\ {0^{T}} & {1}\end{array}\right] \cdot\left[\begin{array}{cc}{A} & {0} \\ {0^{T}} & {1}\end{array}\right]=\left[\begin{array}{cc}{A} & {p} \\ {0^{T}} & {1}\end{array}\right]$

Linear Algebra and Its Applications

Grace H.

Kayleah T.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Kayleah T.

Maria G.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications